Controlling distance
ShiftD duplicating an edge will not cut an edge into the face. The duplicated edge will just hover there, unconnected to the rest of the object. Probably what you want is?...


Select the edge you want to slide along, and CtrlAltSpace create a Custom Orientation from it. (Check 'Use After Creation'). 
Create the edge you want to slide, in the face, any old how, eg.  K cut it with the Knife tool, not worrying about accuracy
GG slide it down all the way to the end of the face, where it will be stopped by the end vertices
GYYYour Number move the edge back again, to the required distance from the end, parallel to the orientation edge.


.. but this method will work just as well with a duplicated edge, if that's what you want.